What caused what? An irreducible account of actual causation

Actual causation is concerned with the question "what caused what?". Consider a transition between two subsequent observations within a system of elements. Even under perfect knowledge of the system, a straightforward answer to this question may not be available. Counterfactual accounts of actual causation based on graphical models, paired with system interventions, have demonstrated initial success in addressing specific problem cases. We present a formal account of actual causation, applicable to discrete dynamical systems of interacting elements, that considers all counterfactual states of a state transition from t-1 to t. Within such a transition, causal links are considered from two complementary points of view: we can ask if any occurrence at time t has an actual cause at t-1, but also if any occurrence at time t-1 has an actual effect at t. We address the problem of identifying such actual causes and actual effects in a principled manner by starting from a set of basic requirements for causation (existence, composition, information, integration, and exclusion). We present a formal framework to implement these requirements based on system manipulations and partitions. This framework is used to provide a complete causal account of the transition by identifying and quantifying the strength of all actual causes and effects linking two occurrences. Finally, we examine several exemplary cases and paradoxes of causation and show that they can be illuminated by the proposed framework for quantifying actual causation.